Modeling Tides with Trigonometry

Date: 03/07/2001 at 21:53:58
From: Saundra Holseth
Subject: Trigonometry, sinusoids

This is a story problem, with multiple parts. I think that if you can 
help me get started, I can finish. 

The depth of water at the end of a pier varies with the tides 
throughout the day. Today, January 23, the high tide occurs at 
4:15 A.M. with a depth of 5.2 meters. The low tide occurs at 
10:27 A.M. with a depth of 2.0 meters.
   
   A. Find a trigonometric equation that models the depth of the water 
      t hours after midnight, and graph it.

   B. Find the depth of the water at noon.

   C. A large boat needs at least 3 meters of water to moor at the end 
      of the pier. During what time period after noon can it safely 
      moor? Show this point on a graph in red.

   D. What is the first time after midnight on February 1 when high 
      tide occurs? Show this point on your graph in blue. (Hint: set 
      your time scale on the x-axis to run from midnight on Jan. 23 to 
      noon on Feb. 1.)

Date: 03/08/2001 at 09:11:10
From: Doctor Wolfson
Subject: Re: Trigonometry, sinusoids

Hi Saundra,

Let's start by writing down exactly what variables we can work with to 
find the answer. The general form of the trig function is:

     y = A cos(Bx + C) + D

I'll explain later why I used cosine instead of sine. Here, A is the 
amplitude from the middle line to the peak, B is the frequency divided 
by 2pi radians, or 360 degrees, C is the horizontal (phase) offset, 
and D is the vertical offset. Now let's find each of these.

A is half the difference between the high tide value, and the low tide 
value, and those numbers are in the problem.

B is related to the frequency. The amount of time it takes to go from 
high tide to low tide is half of one period, so if you subtract those 
times, and double the answer, you'll find the full period. To find the 
value of B, start with 2pi radians (or 360 degrees) and divide by that 
full period; and you'll have B.

C tells us when the graph "starts." I decided to use cosine because if 
C = 0, then it hits "high tide" at x = 0, and we are given the times 
for high tide (instead of "middle tide"). Now you can find what 
fraction of a period after midnight high tide occurs, multiply by 2pi 
(or 360), and write C. Remember that since we want to shift the graph 
to the right, it will have a minus sign.

D is how far above 0 the middle line is. This is just the average of 
high and low tides.

So now you have your trig equation. You can find a noon tide height by 
just evaluating the equation at the right time. For the next part, 
find where the graph crosses y = 3 (or subtract 3 from the function 
and find the zeros.) For the last part, you know that high tide will 
occur after an integer number of periods from the beginning high tide, 
so just find out how many periods it takes to get to Feb. 1.

I hope this helps - feel free to write back if you'd like further 
clarification.

- Doctor Wolfson, The Math Forum
  http://mathforum.org/dr.math/